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A1.1 The quantum mechanics of atoms and moleculesJohn F
Stanton
A1.1.1 INTRODUCTION At the turn of the 19th century, it was
generally believed that the great distance between earth and the
stars would forever limit what could be learned about the universe.
Apart from their approximate size and distance from earth, there
seemed to be no hope of determining intensive properties of stars,
such as temperature and composition. While this pessimistic
attitude may seem quaint from a modern perspective, it should be
remembered that all knowledge gained in these areas has been
obtained by exploiting a scientific technique that did not exist
200 years agospectroscopy. In 1859, Kirchoff made a breakthrough
discovery about the nearest starour sun. It had been known for some
time that a number of narrow dark lines are found when sunlight is
bent through a prism. These absences had been studied
systematically by Fraunhofer, who also noted that dark lines can be
found in the spectrum of other stars; furthermore, many of these
absences are found at the same wavelengths as those in the solar
spectrum. By burning substances in the laboratory, Kirchoff was
able to show that some of the features are due to the presence of
sodium atoms in the solar atmosphere. For the first time, it had
been demonstrated that an element found on our planet is not
unique, but exists elsewhere in the universe. Perhaps most
important, the field of modern spectroscopy was born. Armed with
the empirical knowledge that each element in the periodic table has
a characteristic spectrum, and that heating materials to a
sufficiently high temperature disrupts all interatomic
interactions, Bunsen and Kirchoff invented the spectroscope, an
instrument that atomizes substances in a flame and then records
their emission spectrum. Using this instrument, the elemental
composition of several compounds and minerals were deduced by
measuring the wavelength of radiation that they emit. In addition,
this new science led to the discovery of elements, notably caesium
and rubidium. Despite the enormous benefits of the fledgling field
of spectroscopy for chemistry, the underlying physical processes
were completely unknown a century ago. It was believed that the
characteristic frequencies of elements were caused by (nebulously
defined) vibrations of the atoms, but even a remotely satisfactory
quantitative theory proved to be elusive. In 1885, the Swiss
mathematician Balmer noted that wavelengths in the visible region
of the hydrogen atom emission spectrum could be fitted by the
empirical equation
(A1.1.1)
where m = 2 and n is an integer. Subsequent study showed that
frequencies in other regions of the hydrogen spectrum could be
fitted to this equation by assigning different integer values to m,
albeit with a different value of the constant b. Ritz noted that a
simple modification of Balmers formula(A1.1.2)
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succeeds in fitting all the line spectra corresponding to
different values of m with only the single constant RH. Although
this formula provides an important clue regarding the underlying
processes involved in spectroscopy, more than two decades passed
before a theory of atomic structure succeeded in deriving this
equation from first principles. The origins of line spectra as well
as other unexplained phenomena such as radioactivity and the
intensity profile in the emission spectrum of hot objects
eventually led to a realization that the physics of the day was
incomplete. New ideas were clearly needed before a detailed
understanding of the submicroscopic world of atoms and molecules
could be gained. At the turn of the 20th century, Planck succeeded
in deriving an equation that gave a correct description of the
radiation emitted by an idealized isolated solid (blackbody
radiation). In the derivation, Planck assumed that the energy of
electromagnetic radiation emitted by the vibrating atoms of the
solid cannot have just any energy, but must be an integral multiple
of h, where is the frequency of the radiation and h is now known as
Plancks constant. The resulting formula matched the experimental
blackbody spectrum perfectly. Another phenomenon that could not be
explained by classical physics involved what is now known as the
photoelectric effect. When light impinges on a metal, ionization
leading to ejection of electrons happens only at wavelengths ( =
c/, where c is the speed of light) below a certain threshold. At
shorter wavelengths (higher frequency), the kinetic energy of the
photoelectrons depends linearly on the frequency of the applied
radiation field and is independent of its intensity. These findings
were inconsistent with conventional electromagnetic theory. A
brilliant analysis of this phenomenon by Einstein convincingly
demonstrated that electromagnetic energy is indeed absorbed in
bundles, or quanta (now called photons), each with energy h where h
is precisely the same quantity that appears in Plancks formula for
the blackbody emission spectrum. While the revolutionary ideas of
Planck and Einstein forged the beginnings of the quantum theory,
the physics governing the structure and properties of atoms and
molecules remained unknown. Independent experiments by Thomson,
Weichert and Kaufmann had established that atoms are not the
indivisible entities postulated by Democritus 2000 years ago and
assumed in Daltons atomic theory. Rather, it had become clear that
all atoms contain identical negative charges called electrons. At
first, this was viewed as a rather esoteric feature of matter, the
electron being an entity that would never be of any use to anyone.
With time, however, the importance of the electron and its role in
the structure of atoms came to be understood. Perhaps the most
significant advance was Rutherfords interpretation of the
scattering of alpha particles from a thin gold foil in terms of
atoms containing a very small, dense, positively charged core
surrounded by a cloud of electrons. This picture of atoms is
fundamentally correct, and is now learned each year by millions of
elementary school students. Like the photoelectric effect, the
atomic model developed by Rutherford in 1911 is not consistent with
the classical theory of electromagnetism. In the hydrogen atom, the
force due to Coulomb attraction between the nucleus and the
electron results in acceleration of the electron (Newtons first
law). Classical electromagnetic theory mandates that all
accelerated bodies bearing charge must emit radiation. Since
emission of radiation necessarily results in a loss of energy, the
electron should eventually be captured by the nucleus. But this
catastrophe does not occur. Two years after Rutherfords gold-foil
experiment, the first quantitatively successful theory of an atom
was developed by Bohr. This model was based on a combination of
purely classical ideas, use of Plancks constant h and the bold
assumption that radiative loss of energy does not occur provided
the electron adheres to certain special orbits, or stationary
states. Specifically, electrons that move in a circular path about
the nucleus with a classical angular momentum mvr equal to an
integral multiple of Plancks constant divided by 2 (a quantity of
sufficient general use that it is designated by the simple symbol )
are immune from energy loss in the Bohr model. By simply writing
the classical energy of the orbiting electron in terms of its mass
m, velocity v, distance r from the nucleus and charge e,
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(A1.1.3)
invoking the (again classical) virial theorem that relates the
average kinetic (T) and potential (V) energy of a system governed
by a potential that depends on pairwise interactions of the form rk
via(A1.1.4)
and using Bohrs criterion for stable orbits(A1.1.5)
it is relatively easy to demonstrate that energies associated
with orbits having angular momentum hydrogen atom are given by
in the
(A1.1.6)
with corresponding radii(A1.1.7)
Bohr further postulated that quantum jumps between the different
allowed energy levels are always accompanied by absorption or
emission of a photon, as required by energy conservation,
viz.(A1.1.8)
or perhaps more illustratively(A1.1.9)
precisely the form of the equation deduced by Ritz. The constant
term of equation (A1.1.2) calculated from Bohrs equation did not
exactly reproduce the experimental value at first. However, this
situation was quickly remedied when it was realized that a proper
treatment of the two-particle problem involved use of the reduced
mass of the system mmproton/(m + mproton), a minor modification
that gives striking agreement with experiment.
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Despite its success in reproducing the hydrogen atom spectrum,
the Bohr model of the atom rapidly encountered difficulties.
Advances in the resolution obtained in spectroscopic experiments
had shown that the spectral features of the hydrogen atom are
actually composed of several closely spaced lines; these are not
accounted for by quantum jumps between Bohrs allowed orbits.
However, by modifying the Bohr model to
allow for elliptical orbits and to include the special theory of
relativity, Sommerfeld was able to account for some of the fine
structure of spectral lines. More serious problems arose when the
planetary model was applied to systems that contained more than one
electron. Efforts to calculate the spectrum of helium were
completely unsuccessful, as was a calculation of the spectrum of
the hydrogen molecule ion ( ) that used a generalization of the
Bohr model to treat a problem involving two nuclei. This latter
work formed the basis of the PhD thesis of Pauli, who was to become
one of the principal players in the development of a more mature
and comprehensive theory of atoms and molecules. In retrospect, the
Bohr model of the hydrogen atom contains several flaws. Perhaps
most prominent among these is that the angular momentum of the
hydrogen ground state (n = 1) given by the model is ; it is now
known that the correct value is zero. Efforts to remedy the Bohr
model for its insufficiencies, pursued doggedly by Sommerfeld and
others, were ultimately unsuccessful. This old quantum theory was
replaced in the 1920s by a considerably more abstract framework
that forms the basis for our current understanding of the detailed
physics governing chemical processes. The modern quantum theory,
unlike Bohrs, does not involve classical ideas coupled with an ad
hoc incorporation of Plancks quantum hypothesis. It is instead
founded upon a limited number of fundamental principles that cannot
be proven, but must be regarded as laws of nature. While the modern
theory of quantum mechanics is exceedingly complex and fraught with
certain philosophical paradoxes (which will not be discussed), it
has withstood the test of time; no contradiction between
predictions of the theory and actual atomic or molecular phenomena
has ever been observed. The purpose of this chapter is to provide
an introduction to the basic framework of quantum mechanics, with
an emphasis on aspects that are most relevant for the study of
atoms and molecules. After summarizing the basic principles of the
subject that represent required knowledge for all students of
physical chemistry, the independent-particle approximation so
important in molecular quantum mechanics is introduced. A
significant effort is made to describe this approach in detail and
to communicate how it is used as a foundation for qualitative
understanding and as a basis for more accurate treatments.
Following this, the basic techniques used in accurate calculations
that go beyond the independent-particle picture (variational method
and perturbation theory) are described, with some attention given
to how they are actually used in practical calculations. It is
clearly impossible to present a comprehensive discussion of quantum
mechanics in a chapter of this length. Instead, one is forced to
present cursory overviews of many topics or to limit the scope and
provide a more rigorous treatment of a select group of subjects.
The latter alternative has been followed here. Consequently, many
areas of quantum mechanics are largely ignored. For the most part,
however, the areas lightly touched upon or completely absent from
this chapter are specifically dealt with elsewhere in the
encyclopedia. Notable among these are the interaction between
matter and radiation, spin and magnetism, techniques of quantum
chemistry including the BornOppenheimer approximation, the
HartreeFock method and electron correlation, scattering theory and
the treatment of internal nuclear motion (rotation and vibration)
in molecules.
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A1.1.2 CONCEPTS OF QUANTUM MECHANICSA1.1.2.1 BEGINNINGS AND
FUNDAMENTAL POSTULATES
The modern quantum theory derives from work done independently
by Heisenberg and Schrdinger in the mid-1920s. Superficially, the
mathematical formalisms developed by these individuals appear very
different; the quantum mechanics of Heisenberg is based on the
properties of matrices, while that of Schrdinger is founded upon a
differential equation that bears similarities to those used in the
classical theory of waves. Schrdingers formulation was strongly
influenced by the work of de Broglie, who made the
revolutionary
hypothesis that entities previously thought to be strictly
particle-like (electrons) can exhibit wavelike behaviour (such as
diffraction) with particle wavelength and momentum (p) related by
the equation = h/p. This truly startling premise was subsequently
verified independently by Davisson and Germer as well as by
Thomson, who showed that electrons exhibit diffraction patterns
when passed through crystals and very small circular apertures,
respectively. Both the treatment of Heisenberg, which did not make
use of wave theory concepts, and that of Schrdinger were
successfully applied to the calculation of the hydrogen atom
spectrum. It was ultimately proven by both Pauli and Schrdinger
that the matrix mechanics of Heisenberg and the wave mechanics of
Schrdinger are mathematically equivalent. Connections between the
two methods were further clarified by the transformation theory of
Dirac and Jordan. The importance of this new quantum theory was
recognized immediately and Heisenberg, Schrdinger and Dirac shared
the 1932 Nobel Prize in physics for their work. While not unique,
the Schrdinger picture of quantum mechanics is the most familiar to
chemists principally because it has proven to be the simplest to
use in practical calculations. Hence, the remainder of this section
will focus on the Schrdinger formulation and its associated
wavefunctions, operators and eigenvalues. Moreover, effects
associated with the special theory of relativity (which include
spin) will be ignored in this subsection. Treatments of alternative
formulations of quantum mechanics and discussions of relativistic
effects can be found in the reading list that accompanies this
chapter. Like the geometry of Euclid and the mechanics of Newton,
quantum mechanics is an axiomatic subject. By making several
assertions, or postulates, about the mathematical properties of and
physical interpretation associated with solutions to the Schrdinger
equation, the subject of quantum mechanics can be applied to
understand behaviour in atomic and molecular systems. The first of
these postulates is: 1. Corresponding to any collection of n
particles, there exists a time-dependent function (q1, q2, . . .,
qn; t) that comprises all information that can be known about the
system. This function must be continuous and single valued, and
have continuous first derivatives at all points where the classical
force has a finite magnitude. In classical mechanics, the state of
the system may be completely specified by the set of Cartesian
particle coordinates ri and velocities dri/dt at any given time.
These evolve according to Newtons equations of motion. In
principle, one can write down equations involving the state
variables and forces acting on the particles which can be solved to
give the location and velocity of each particle at any later (or
earlier) time t, provided one knows the precise state of the
classical system at time t. In quantum mechanics, the state of the
system at time t is instead described by a well behaved
mathematical function of the particle coordinates qi rather than a
simple list of positions and velocities.
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The relationship between this wavefunction (sometimes called
state function) and the location of particles in the system forms
the basis for a second postulate: 2. The product of (q1, q2, . . .,
qn; t) and its complex conjugate has the following physical
interpretation. The probability of finding the n particles of the
system in the regions bounded by the coordinates and at time t is
proportional to the integral
(A1.1.10)
The proportionality between the integral and the probability can
be replaced by an equivalence if the wavefunction is scaled
appropriately. Specifically, since the probability that the n
particles will be found somewhere must be unity, the wavefunction
can be scaled so that the equality(A1.1.11)
is satisfied. The symbol d introduced here and used throughout
the remainder of this section indicates that the integral is to be
taken over the full range of all particle coordinates. Any
wavefunction that satisfies equation (A1.1.11) is said to be
normalized. The product * corresponding to a normalized
wavefunction is sometimes called a probability, but this is an
imprecise use of the word. It is instead a probability density,
which must be integrated to find the chance that a given
measurement will find the particles in a certain region of space.
This distinction can be understood by considering the classical
counterpart of * for a single particle moving on the x-axis. In
classical mechanics, the probability at time t for finding the
particle at the coordinate (x) obtained by propagating Newtons
equations of motion from some set of initial conditions is exactly
equal to one; it is zero for any other value of x. What is the
corresponding probability density function, P(x; t) Clearly, P(x;
t) vanishes at all points other than x since its integral over any
interval that does not include x must equal zero. At x, the value
of P(x; t) must be chosen so that the normalization
condition(A1.1.12)
is satisfied. Functions such as this play a useful role in
quantum mechanics. They are known as Dirac delta functions, and are
designated by (r r0). These functions have the properties(A1.1.13)
(A1.1.14) (A1.1.15)
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Although a seemingly odd mathematical entity, it is not hard to
appreciate that a simple one-dimensional realization of the
classical P(x; t) can be constructed from the familiar Gaussian
distribution centred about x by letting the standard deviation ()
go to zero,(A1.1.16)
Hence, although the probability for finding the particle at x is
equal to one, the corresponding probability density function is
infinitely large. In quantum mechanics, the probability density is
generally nonzero for all values of the coordinates, and its
magnitude can be used to determine which regions are most likely to
contain particles. However, because the number of possible
coordinates is infinite, the probability associated with any
precisely specified choice is zero. The discussion above shows a
clear distinction between classical and quantum mechanics; given a
set of initial conditions, the locations of the particles are
determined exactly at all future times in the former, while one
generally can speak only about the probability associated with a
given range of coordinates in quantum mechanics.
To extract information from the wavefunction about properties
other than the probability density, additional postulates are
needed. All of these rely upon the mathematical concepts of
operators, eigenvalues and eigenfunctions. An extensive discussion
of these important elements of the formalism of quantum mechanics
is precluded by space limitations. For further details, the reader
is referred to the reading list supplied at the end of this
chapter. In quantum mechanics, the classical notions of position,
momentum, energy etc are replaced by mathematical operators that
act upon the wavefunction to provide information about the system.
The third postulate relates to certain properties of these
operators: 3. Associated with each system property A is a linear,
Hermitian operator . Although not a unique prescription, the
quantum-mechanical operators can be obtained from their classical
counterparts A by making the substitutions x x (coordinates); t t
(time); pq -i /q (component of momentum). Hence, the
quantum-mechanical operators of greatest relevance to the dynamics
of an n-particle system such as an atom or molecule are:
Dynamical variable A
Classical quantity
Quantum-mechanical operator
Time Position of particle i Momentum of particle i Angular
momentum of particle i Kinetic energy of particle i Potential
energy
t ri mivi mivi ri
t ri i i i i ri
V(q, t)
V(q, t)
-8-
where the gradient
(A1.1.17)
and Laplacian
(A1.1.18)
operators have been introduced. Note that a potential energy
which depends upon only particle coordinates and time has exactly
the same form in classical and quantum mechanics. A particularly
useful operator in quantum mechanics is that which corresponds to
the total energy. This Hamiltonian operator is obtained by simply
adding the potential and kinetic energy operators
(A1.1.19)
The relationship between the abstract quantum-mechanical
operators and the corresponding physical quantities A is the
subject of the fourth postulate, which states: 4. If the system
property A is measured, the only values that can possibly be
observed are those that correspond to eigenvalues of the
quantum-mechanical operator . An illustrative example is provided
by investigating the possible momenta for a single particle
travelling in the x-direction, px. First, one writes the equation
that defines the eigenvalue condition(A1.1.20)
where is an eigenvalue of the momentum operator and f(x) is the
associated eigenfunction. It is easily verified that this
differential equation has an infinite number of solutions of the
form(A1.1.21)
with corresponding eigenvalues(A1.1.22)
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in which k can assume any value. Hence, nature places no
restrictions on allowed values of the linear momentum. Does this
mean that a quantum-mechanical particle in a particular state (x;
t) is allowed to have any value of px? The answer to this question
is yes, but the interpretation of its consequences rather subtle.
Eventually a fifth postulate will be required to establish the
connection between the quantum-mechanical wavefunction and the
possible outcomes associated with measuring properties of the
system. It turns out that the set of possible momenta for our
particle depends entirely on its wavefunction, as might be expected
from the first postulate given above. The infinite set of solutions
to equation (A1.1.20) means only that no values of the momentum are
excluded, in the sense that they can be associated with a particle
described by an appropriately chosen wavefunction. However, the
choice of a specific function might (or might not) impose
restrictions on which values of px are allowed. The rather
complicated issues raised in the preceding paragraph are central to
the subject of quantum mechanics, and their resolution forms the
basis of one of the most important postulates associated with the
Schrdinger formulation of the subject. In the example above,
discussion focuses entirely on the eigenvalues of the momentum
operator. What significance, if any, can be attached to the
eigenfunctions of quantummechanical operators? In the interest of
simplicity, the remainder of this subsection will focus entirely on
the quantum mechanics associated with operators that have a finite
number of eigenvalues. These are said to have a discrete spectrum,
in contrast to those such as the linear momentum, which have a
continuous spectrum. Discrete spectra of eigenvalues arise whenever
boundaries limit the region of space in which a system can be.
Examples are particles in hard-walled boxes, or soft-walled shells
and particles attached to springs. The results developed below can
all be generalized to the continuous case, but at the expense of
increased mathematical complexity. Readers interested in these
details should consult chapter 1 of Landau and Lifschitz (see
additional reading).
It can be shown that the eigenfunctions of Hermitian operators
necessarily exhibit a number of useful mathematical properties.
First, if all eigenvalues are distinct, the set of eigenfunctions
{f1, f2 fn} are orthogonal in the sense that the integral of the
product formed from the complex conjugate of eigenfunction and
eigenfunction k (fk) vanishes unless j = k,
(A1.1.23)
If there are identical eigenvalues (a common occurrence in
atomic and molecular quantum mechanics), it is permissible to form
linear combinations of the eigenfunctions corresponding to these
degenerate eigenvalues, as these must also be eigenfunctions of the
operator. By making a judicious choice of the expansion
coefficients, the degenerate eigenfunctions can also be made
orthogonal to one another. Another useful property is that the set
of eigenfunctions is said to be complete. This means that any
function of the coordinates that appear in the operator can be
written as a linear combination of its eigenfunctions, provided
that the function obeys the same boundary conditions as the
eigenfunctions and shares any fundamental symmetry property that is
common to all of them. If, for example, all of the eigenfunctions
vanish at some point in space, then only functions that vanish at
the same point can be written as linear combinations of the
eigenfunctions. Similarly, if the eigenfunctions of a particular
operator in one dimension are all odd functions of the coordinate,
then all linear combinations of them must also be odd. It is
clearly impossible in the latter case to expand functions such as
cos(x), exp(x2) etc in terms of odd functions. This qualification
is omitted in some elementary treatments of quantum mechanics, but
it is one that turns out to be important for systems containing
several identical particles. Nevertheless, if these criteria are
met by a suitable function g, then it is always possible to find
coefficients ck such that
-10(A1.1.24)
where the coefficient cj is given by
(A1.1.25)
If the eigenfunctions are normalized, this expression reduces
to(A1.1.26)
When normalized, the eigenfunctions corresponding to a Hermitian
operator are said to represent an orthonormal set. The mathematical
properties discussed above are central to the next postulate: 5. In
any experiment, the probability of observing a particular
non-degenerate value for the system property A can be determined by
the following procedure. First, expand the wavefunction in terms of
the complete set of normalized eigenfunctions of the
quantummechanical operator, ,
(A1.1.27)
The probability of measuring A = k, where k is the eigenvalue
associated with the normalized eigenfunction k, is precisely equal
to . For degenerate eigenvalues, the probability of observation is
given by ck2, where the sum is taken over all of the eigenfunctions
k that correspond to the degenerate eigenvalue k. At this point, it
is appropriate to mention an elementary concept from the theory of
probability. If there are n possible numerical outcomes (n)
associated with a particular process, the average value can be
calculated by summing up all of the outcomes, each weighted by its
corresponding probability(A1.1.28)
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As an example, the possible outcomes and associated
probabilities for rolling a pair of six-sided dice are
Sum
Probability
2 3 4 5 6 7 8 9 10 11 12
1/36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36
The average value is therefore given by the sum . What does this
have to do with quantum mechanics? To establish a connection, it is
necessary to first expand the wavefunction in terms of the
eigenfunctions of a quantum-mechanical operator ,
(A1.1.29)
We will assume that both the wavefunction and the orthogonal
eigenfunctions are normalized, which implies that
(A1.1.30)
Now, the operator is applied to both sides of equation
(A1.1.29), which because of its linearity, gives(A1.1.31)
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where k represents the eigenvalue associated with the
eigenfunction k. Next, both sides of the preceding equation are
multiplied from the left by the complex conjugate of the
wavefunction and integrated over all space
(A1.1.32)
(A1.1.33) (A1.1.34)
The last identity follows from the orthogonality property of
eigenfunctions and the assumption of normalization. The right-hand
side in the final result is simply equal to the sum over all
eigenvalues of the operator (possible results of the measurement)
multiplied by the respective probabilities. Hence, an important
corollary to the fifth postulate is established:(A1.1.35)
This provides a recipe for calculating the average value of the
system property associated with the quantummechanical operator ,
for a specific but arbitrary choice of the wavefunction , notably
those choices which are not eigenfunctions of . The fifth postulate
and its corollary are extremely important concepts. Unlike
classical mechanics, where everything can in principle be known
with precision, one can generally talk only about the probabilities
associated with each member of a set of possible outcomes in
quantum mechanics. By making a measurement of the quantity A, all
that can be said with certainty is that one of the eigenvalues of
will be observed, and its probability can be calculated precisely.
However, if it happens that the wavefunction corresponds to one of
the eigenfunctions of the operator , then and only then is the
outcome of the experiment certain: the measured value of A will be
the corresponding eigenvalue. Up until now, little has been said
about time. In classical mechanics, complete knowledge about the
system at any time t suffices to predict with absolute certainty
the properties of the system at any other time t. The situation is
quite different in quantum mechanics, however, as it is not
possible to know everything about the system at any time t.
Nevertheless, the temporal behavior of a quantum-mechanical system
evolves in a well defined way that depends on the Hamiltonian
operator and the wavefunction according to the last postulate 6.
The time evolution of the wavefunction is described by the
differential equation
(A1.1.36)
The differential equation above is known as the time-dependent
Schrdinger equation. There is an interesting and
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intimate connection between this equation and the classical
expression for a travelling wave
(A1.1.37)
To convert (A1.1.37) into a quantum-mechanical form that
describes the matter wave associated with a free particle
travelling through space, one might be tempted to simply make the
substitutions = E/h (Plancks hypothesis) and = h/p (de Broglies
hypothesis). It is relatively easy to verify that the resulting
expression satisfies the time-dependent Schrdinger equation.
However, it should be emphasized that this is not a derivation, as
there is no compelling reason to believe that this ad hoc procedure
should yield one of the fundamental equations of physics. Indeed,
the time-dependent Schrdinger equation cannot be derived in a
rigorous way and therefore must be regarded as a postulate. The
time-dependent Schrdinger equation allows the precise determination
of the wavefunction at any time t from knowledge of the
wavefunction at some initial time, provided that the forces acting
within the system are known (these are required to construct the
Hamiltonian). While this suggests that quantum mechanics has a
deterministic component, it must be emphasized that it is not the
observable system properties that evolve in a precisely specified
way, but rather the probabilities associated with values that might
be found for them in a measurement.A1.1.2.2 STATIONARY STATES,
SUPERPOSITION AND UNCERTAINTY
From the very beginning of the 20th century, the concept of
energy conservation has made it abundantly clear that
electromagnetic energy emitted from and absorbed by material
substances must be accompanied by compensating energy changes
within the material. Hence, the discrete nature of atomic line
spectra suggested that only certain energies are allowed by nature
for each kind of atom. The wavelengths of radiation emitted or
absorbed must therefore be related to the difference between energy
levels via Plancks hypothesis, E = h = hc/. The Schrdinger picture
of quantum mechanics summarized in the previous subsection allows
an important deduction to be made that bears directly on the
subject of energy levels and spectroscopy. Specifically, the
energies of spectroscopic transitions must correspond precisely to
differences between distinct eigenvalues of the Hamiltonian
operator, as these correspond to the allowed energy levels of the
system. Hence, the set of eigenvalues of the Hamiltonian operator
are of central importance in chemistry. These can be determined by
solving the so-called time-independent Schrdinger
equation,(A1.1.38)
for the eigenvalues Ek and eigenfunctions k. It should be clear
that the set of eigenfunctions and eigenvalues does not evolve with
time provided the Hamiltonian operator itself is time independent.
Moreover, since the
eigenfunctions of the Hamiltonian (like those of any other
operator) form a complete set, it is always possible to expand the
exact wavefunction of the system at any time in terms of
them:(A1.1.39)
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It is important to point out that this expansion is valid even
if time-dependent terms are added to the Hamiltonian (as, for
example, when an electric field is turned on). If there is more
than one nonzero value of cj at any time t, then the system is said
to be in a superposition of the energy eigenstates k associated
with non-vanishing expansion coefficients, ck. If it were possible
to measure energies directly, then the fifth postulate of the
previous section tells us that the probability of finding energy Ek
in a given measurement would be . When a molecule is isolated from
external fields, the Hamiltonian contains only kinetic energy
operators for all of the electrons and nuclei as well as terms that
account for repulsion and attraction between all distinct pairs of
like and unlike charges, respectively. In such a case, the
Hamiltonian is constant in time. When this condition is satisfied,
the representation of the time-dependent wavefunction as a
superposition of Hamiltonian eigenfunctions can be used to
determine the time dependence of the expansion coefficients. If
equation (A1.1.39) is substituted into the time-dependent
Schrdinger equation
(A1.1.40)
the simplification(A1.1.41)
can be made to the right-hand side since the restriction of a
time-independent Hamiltonian means that k is always an
eigenfunction of H. By simply equating the coefficients of the k,
it is easy to show that the choice(A1.1.42)
for the time-dependent expansion coefficients satisfies equation
(A1.1.41). Like any differential equation, there are an infinite
number of solutions from which a choice must be made to satisfy
some set of initial conditions. The state of the quantum-mechanical
system at time t = 0 is used to fix the arbitrary multipliers ck
(0), which can always be chosen as real numbers. Hence, the
wavefunction becomes
(A1.1.43)
Suppose that the system property A is of interest, and that it
corresponds to the quantum-mechanical operator . The average value
of A obtained in a series of measurements can be calculated by
exploiting the corollary to the fifth postulate
(A1.1.44)
-15-
Now consider the case where is itself a time-independent
operator, such as that for the position, momentum or angular
momentum of a particle or even the energy of the benzene molecule.
In these cases, the timedependent expansion coefficients are
unaffected by application of the operator, and one obtains
(A1.1.45)
As one might expect, the first term that contributes to the
expectation value of A is simply its value at t = 0, while the
second term exhibits an oscillatory time dependence. If the
superposition initially includes large contributions from states of
widely varying energy, then the oscillations in A will be rapid. If
the states that are strongly mixed have similar energies, then the
timescale for oscillation in the properties will be slower.
However, there is one special class of system properties A that
exhibit no time dependence whatsoever. If (and only if) every one
of the states k is an eigenfunction of , then the property of
orthogonality can be used to show that every contribution to the
second term vanishes. An obvious example is the Hamiltonian
operator itself; it turns out that the expectation value for the
energy of a system subjected to forces that do not vary with time
is a constant. Are there other operators that share the same set of
eigenfunctions k with , and if so, how can they be recognized? It
can be shown that any two operators which satisfy the
property(A1.1.46)
for all functions f share a common set of eigenfunctions, and A
and B are said to commute. (The symbol [ , ] meaning , is called
the commutator of the operators and .) Hence, there is no time
dependence for the expectation value of any system property that
corresponds to a quantum-mechanical operator that commutes with the
Hamiltonian. Accordingly, these quantities are known as constants
of the motion: their average values will not vary, provided the
environment of the system does not change (as it would, for
example, if an electromagnetic field were suddenly turned on). In
nonrelativistic quantum mechanics, two examples of constants of the
motion are the square of the total angular momentum, as well as its
projection along an arbitrarily chosen axis. Other operators, such
as that for the dipole moment, do not commute with the Hamiltonian
and the expectation value associated with the corresponding
properties can indeed oscillate with time. It is important to note
that the frequency of these oscillations is given by differences
between the allowed energies of the system divided by Plancks
constant. These are the so-called Bohr frequencies, and it is
perhaps not surprising that these are exactly the frequencies of
electromagnetic radiation that cause transitions between the
corresponding energy levels. Close inspection of equation (A1.1.45)
reveals that, under very special circumstances, the expectation
value does not change with time for any system properties that
correspond to fixed (static) operator representations.
Specifically, if the spatial part of the time-dependent
wavefunction is the exact eigenfunction j of the Hamiltonian, then
cj(0) = 1 (the zero of time can be chosen arbitrarily) and all
other ck(0) = 0. The second term clearly vanishes in these cases,
which are known as stationary states. As the name implies, all
observable properties of these states do not vary with time. In a
stationary state, the energy of the system has a precise value (the
corresponding eigenvalue of ) as do observables that are associated
with operators that commute with . For all other properties (such
as the position and momentum),
-16-
one can speak only about average values or probabilities
associated with a given measurement, but these quantities
themselves do not depend on time. When an external perturbation
such as an electric field is applied or a collision with another
atom or molecule occurs, however, the system and its properties
generally will evolve with time. The energies that can be absorbed
or emitted in these processes correspond precisely to differences
in the stationary state energies, so it should be clear that
solving the time-independent Schrdinger equation for the stationary
state wavefunctions and eigenvalues provides a wealth of
spectroscopic information. The importance of stationary state
solutions is so great that it is common to refer to equation
(A1.1.38) as the Schrdinger equation, while the qualified name
time-dependent Schrdinger equation is generally used for equation
(A1.1.36). Indeed, the subsequent subsections are devoted entirely
to discussions that centre on the former and its exact and
approximate solutions, and the qualifier time independent will be
omitted. Starting with the quantum-mechanical postulate regarding a
one-to-one correspondence between system properties and Hermitian
operators, and the mathematical result that only operators which
commute have a common set of eigenfunctions, a rather remarkable
property of nature can be demonstrated. Suppose that one desires to
determine the values of the two quantities A and B, and that the
corresponding quantum-mechanical operators do not commute. In
addition, the properties are to be measured simultaneously so that
both reflect the same quantum-mechanical state of the system. If
the wavefunction is neither an eigenfunction of nor , then there is
necessarily some uncertainty associated with the measurement. To
see this, simply expand the wavefunction in terms of the
eigenfunctions of the relevant operators
(A1.1.47) (A1.1.48)
where the eigenfunctions and of operators and , respectively,
are associated with corresponding eigenvalues and . Given that is
not an eigenfunction of either operator, at least two of the
coefficients ak and two of the bk must be nonzero. Since the
probability of observing a particular eigenvalue is proportional to
the square of the expansion coefficient corresponding to the
associated eigenfunction, there will be no less than four possible
outcomes for the set of values A and B. Clearly, they both cannot
be determined precisely. Indeed, under these conditions, neither of
them can be! In a more favourable case, the wavefunction might
indeed correspond to an eigenfunction of one of the operators. If ,
then a measurement of A necessarily yields , and this is an
unambiguous result. What can be said about the measurement of B in
this case? It has already been said that the eigenfunctions of two
commuting operators are identical, but here the pertinent issue
concerns eigenfunctions of two operators that do not commute.
Suppose is an eigenfunction of . Then, it must be true that
(A1.1.49)
-17-
If is also an eigenfunction of , then it follows that , which
contradicts the assumption that and do not commute. Hence, no
nontrivial eigenfunction of can also be an eigenfunction of .
Therefore, if measurement of A yields a precise result, then some
uncertainty must be associated with B. That is, the expansion of in
terms of eigenfunctions of (equation (A1.1.48)) must have at least
two nonvanishing coefficients; the corresponding eigenvalues
therefore represent distinct possible outcomes of the experiment,
each having probability . A physical interpretation of is the
process of measuring the value of A for a system in a state with a
unique value for this property . However represents a measurement
that changes the state of the system, so that if after we measure B
and then measure A, we would no longer find as its value: . The
Heisenberg uncertainty principle offers a rigorous treatment of the
qualitative picture sketched above. If several measurements of A
and B are made for a system in a particular quantum state, then
quantitative uncertainties are provided by standard deviations in
the corresponding measurements. Denoting these as A and B,
respectively, it can be shown that
(A1.1.50)
One feature of this inequality warrants special attention. In
the previous paragraph it was shown that the precise measurement of
A made possible when is an eigenfunction of necessarily results in
some uncertainty in a simultaneous measurement of B when the
operators and do not commute. However, the mathematical statement
of the uncertainty principle tells us that measurement of B is in
fact completely uncertain: one can say nothing at all about B apart
from the fact that any and all values of B are equally probable! A
specific example is provided by associating A and B with the
position and momentum of a . If particle moving along the x-axis.
It is rather easy to demonstrate that [px, x] = i , so that the
system happens to be described by a Dirac delta function at the
point x0 (which is an eigenfunction of the position operator
corresponding to eigenvalue x0), then the probabilities associated
with possible momenta can be determined by expanding (xx0) in terms
of the momentum eigenfunctions A exp(ikx). Carrying out such a
calculation shows that all of the infinite number of possible
momenta (the momentum operator has a continuous spectrum) appear in
the wavefunction expansion, all with precisely the same weight.
Hence, no particular momentum or (more properly in this case) range
bounded by px + dpx is more likely to be observed than any
other.A1.1.2.3 SOME QUALITATIVE FEATURES OF STATIONARY STATES
A great number of qualitative features associated with the
stationary states that correspond to solutions of the
time-independent Schrdinger can be worked out from rather general
mathematical considerations and use of the postulates of quantum
mechanics. Mastering these concepts and the qualifications that may
apply to them is essential if one is to obtain an intuitive feeling
for the subject. In general, the systems of interest to chemists
are atoms and molecules, both in isolation as well as how they
interact with each other or with an externally applied field. In
all of these cases, the forces acting upon the particles in the
system give rise to a potential energy function that varies with
the positions of the particles, strength of the applied fields etc.
In general, the potential is a smoothly varying function of the
coordinates, either growing without bound for large values of the
coordinates or tending asymptotically towards a finite value. In
these cases, there is necessarily a minimum value at what is known
as the global equilibrium position (there may be several global
minima that are equivalent by symmetry). In many cases, there are
also other minima
-18-
(meaning that the matrix of second derivatives with respect to
the coordinates has only non-negative eigenvalues) that have higher
energies, which are called local minima. If the potential becomes
infinitely large for infinite values of the coordinates (as it
does, for example, when the force on a particle varies linearly
with its displacement from equilibrium) then all solutions to the
Schrdinger equation are known as bound states; that with the
smallest eigenvalue is called the ground state while the others are
called excited states. In other cases, such as potential functions
that represent realistic models for diatomic molecules by
approaching a constant finite value at large separation (zero force
on the particles, with a finite dissociation energy), there are two
classes of solutions. Those associated with eigenvalues that are
below the asymptotic value of the potential energy are the bound
states, of which there is usually a finite number; those having
higher energies are called the scattering (or continuum) states and
form a continuous spectrum. The latter are dealt with in section
A3.11 of the encyclopedia and will be mentioned here only when
necessary for mathematical reasons. Bound state solutions to the
Schrdinger equation decay to zero for infinite values of the
coordinates, and are therefore integrable since they are continuous
functions in accordance with the first postulate. The solutions may
assume zero values elsewhere in space and these regionswhich may be
a point, a plane or a three- or higher-dimensional hypersurfaceare
known as nodes. From the mathematical theory of differential
eigenvalue equations, it can be demonstrated that the lowest
eigenvalue is always associated with an eigenfunction that has the
same sign at all points in space. From this result, which can be
derived from the calculus of variations, it follows that the
wavefunction corresponding to the smallest eigenvalue of the
Hamiltonian must have no nodes. It turns out, however, that
relativistic considerations require that this statement be
qualified. For systems that contain more than two identical
particles of a specific type, not all solutions to the Schrdinger
equation are allowed by nature. Because of this restriction, which
is described in subsection (A1.1.3.3) , it turns out that the
ground states of lithium, all larger atoms and all molecules other
than , H2 and isoelectronic species have nodes. Nevertheless, our
conceptual understanding of electronic structure as well as the
basis for almost all highly accurate calculations is ultimately
rooted in a single-particle approximation. The quantum mechanics of
one-particle systems is therefore important in chemistry. Shapes of
the ground- and first three excited-state wavefunctions are shown
in figure A1.1.1 for a particle in one dimension subject to the
potential , which corresponds to the case where the force acting on
the particle is proportional in magnitude and opposite in direction
to its displacement from equilibrium (f V = kx). The corresponding
Schrdinger equation
(A1.1.51)
can be solved analytically, and this problem (probably familiar
to most readers) is that of the quantum harmonic oscillator. As
expected, the ground-state wavefunction has no nodes. The first
excited state has a single node, the second two nodes and so on,
with the number of nodes growing with increasing magnitude of the
eigenvalue. From the form of the kinetic energy operator, one can
infer that regions where the slope of the wavefunction is changing
rapidly (large second derivatives) are associated with large
kinetic energy. It is quite reasonable to accept that wavefunctions
with regions of large curvature (where the function itself has
appreciable magnitude) describe states with high energy, an
expectation that can be made rigorous by applying a
quantum-mechanical version of the virial theorem.
-19-
Figure A1.1.1. Wavefunctions for the four lowest states of the
harmonic oscillator, ordered from the n = 0 ground state (at the
bottom) to the n = 3 state (at the top). The vertical displacement
of the plots is chosen so that the location of the classical
turning points are those that coincide with the superimposed
potential function (dotted line). Note that the number of nodes in
each state corresponds to the associated quantum number.
Classically, a particle with fixed energy E described by a
quadratic potential will move back and forth between the points
where V = E, known as the classical turning points. Movement beyond
the classical turning points is forbidden, because energy
conservation implies that the particle will have a negative kinetic
energy in these regions, and imaginary velocities are clearly
inconsistent with the Newtonian picture of the universe. Inside the
turning points, the particle will have its maximum kinetic energy
as it passes through the minimum, slowing in its climb until it
comes to rest and subsequently changes direction at the turning
points (imagine a marble rolling in a parabola). Therefore, if a
camera were to take snapshots of the particle at random intervals,
most of the pictures would show the particle near the turning
points (the equilibrium position is actually the least likely
location for the particle). A more detailed analysis of the problem
shows that the probability of seeing the classical particle in the
neighbourhood of a given position x is proportional to . Note that
the situation found for the ground state described by quantum
mechanics bears very little resemblance to the classical situation.
The particle is most likely to be found at the equilibrium position
and, within the classically allowed region, least likely to be seen
at the turning points. However, the situation is even stranger than
this: the probability of finding the particle outside the turning
points is non-zero! This phenomenon, known as tunnelling, is not
unique to the harmonic oscillator. Indeed, it occurs for bound
states described by every potential
-20-
that tends asymptotically to a finite value since the
wavefunction and its derivatives must approach zero in a
smooth fashion for large values of the coordinates where (by the
definition of a bound state) V must exceed E. However, at large
energies (see the 29th excited state probability density in figure
A1.1.2, the situation is more consistent with expectations based on
classical theory: the probability density has its largest value
near the turning points, the general appearance is as implied by
the classical formula (if one ignores the oscillations) and its
magnitude in the classically forbidden region is reduced
dramatically with respect to that found for the low-lying states.
This merging of the quantum-mechanical picture with expectations
based on classical theory always occurs for highly excited states
and is the basis of the correspondence principle.
Figure A1.1.2. Probability density (*) for the n = 29 state of
the harmonic oscillator. The vertical state is chosen as in figure
A1.1.1, so that the locations of the turning points coincide with
the superimposed potential function. The energy level spectrum of
the harmonic oscillator is completely regular. The ground state
energy is given by h, where is the classical frequency of
oscillation given by
(A1.1.52)
-21-
although it must be emphasized that our inspection of the
wavefunction shows that the motion of the particle cannot be
literally thought of in this way. The energy of the first excited
state is h above that of the ground state and precisely the same
difference separates each excited state from those immediately
above and below. A different example is provided by a particle
trapped in the Morse potential
(A1.1.53)
originally suggested as a realistic model for the vibrational
motion of diatomic molecules. Although the wavefunctions associated
with the Morse levels exhibit largely the same qualitative features
as the harmonic oscillator functions and are not shown here, the
energy level structures associated with the two systems are
qualitatively different. Since V(x) tends to a finite value (De)
for large x, there are only a limited number of bound state
solutions, and the spacing between them decreases with increasing
eigenvalue. This is another general feature; energy level spacings
for states associated with potentials that tend towards asymptotic
values at infinity tend to decrease with increasing quantum number.
The one-dimensional cases discussed above illustrate many of the
qualitative features of quantum mechanics, and their relative
simplicity makes them quite easy to study. Motion in more than one
dimension and (especially) that of more than one particle is
considerably more complicated, but many of the general features of
these systems can be understood from simple considerations. While
one relatively common feature of multidimensional problems in
quantum mechanics is degeneracy, it turns out that the ground state
must be non-degenerate. To prove this, simply assume the opposite
to be true, i.e.(A1.1.54) (A1.1.55)
where E0 is the ground state energy, and(A1.1.56)
In order to satisfy equation (A1.1.56), the two functions must
have identical signs at some points in space and different signs
elsewhere. It follows that at least one of them must have at least
one node. However, this is incompatible with the nodeless property
of ground-state eigenfunctions. Having established that the ground
state of a single-particle system is non-degenerate and nodeless,
it is straightforward to prove that the wavefunctions associated
with every excited state must contain at least one node (though
they need not be degenerate!), just as seen in the example
problems. It follows from the orthogonality of eigenfunctions
corresponding to a Hermitian operator that
(A1.1.57)
-22-
for all excited states x. In order for this equality to be
satisfied, it is necessary that the integrand either vanishes at
all points in space (which contradicts the assumption that both g
and x are nodeless) or is positive in some regions of space and
negative in others. Given that the ground state has no nodes, the
latter condition can be satisfied only if the excited-state
wavefunction changes sign at one or more points in space. Since the
first postulate states that all wavefunctions are continuous, it is
therefore necessary that x has at least one node. In classical
mechanics, it is certainly possible for a system subject to
dissipative forces such as friction to come to rest. For example, a
marble rolling in a parabola lined with sandpaper will eventually
lose its kinetic energy and come to rest at the bottom. Rather
remarkably, making a measurement of E that coincides with
Vmin (as would be found classically for our stationary marble)
is incompatible with quantum mechanics. Turning back to our
example, the ground-state energy is indeed larger than the minimum
value of the potential energy for the harmonic oscillator. That
this property of zero-point energy is guaranteed in quantum
mechanics can be demonstrated by straightforward application of the
basic principles of the subject. Unlike nodal features of the
wavefunction, the arguments developed here also hold for
many-particle systems. Suppose the total energy of a stationary
state is E. Since the energy is the sum of kinetic and potential
energies, it must be true that expectation values of the kinetic
and potential energies are related according to
(A1.1.58)
If the total energy associated with the state is equal to the
potential energy at the equilibrium position, it follows
that(A1.1.59)
Two cases must be considered. In the first, it will be assumed
that the wavefunction is nonzero at one or more points for which V
> Vmin (for the physically relevant case of a smoothly varying
and continuous potential, this includes all possibilities other
than that in which the wavefunction is a Dirac delta function at
the equilibrium position). This means that V must also be greater
than Vmin thereby forcing the average kinetic energy to be
negative. This is not possible. The kinetic energy operator for a
quantum-mechanical particle moving in the x-direction has the
(unnormalized) eigenfunctions(A1.1.60)
where
(A1.1.61)
and are the corresponding eigenvalues. It can be seen that
negative values of give rise to real arguments of the exponential
and correspondingly divergent eigenfunctions. Zero and non-negative
values are associated with constant and oscillatory solutions in
which the argument of the exponential vanishes or is imaginary,
respectively. Since divergence of the actual wavefunction is
incompatible with its probabilistic interpretation, no contribution
from negative eigenfunctions can appear when the wavefunction is
expanded in terms of kinetic energy eigenfunctions.
-23-
It follows from the fifth postulate that the kinetic energy of
each particle in the system (and therefore the total kinetic
energy) is restricted to non-negative values. Therefore, the
expectation value of the kinetic energy cannot be negative. The
other possibility is that the wavefunction is non-vanishing only
when V = Vmin. For the case of a smoothly varying, continuous
potential, this corresponds to a state described by a Dirac delta
function at the equilibrium position, which is the
quantum-mechanical equivalent of a particle at rest. In any event,
the fact that the wavefunction vanishes at all points for which V
Vmin means that the expectation value of the kinetic energy
operator must also vanish if there is to be no zeropoint energy.
Considering the discussion above, this can occur only when the
wavefunction is the same as the zero-kinetic-energy eigenfunction (
= constant). This contradicts the assumption used in this case,
where the wavefunction is a
delta function. Following the general arguments used in both
cases above, it is easily shown that E can only be larger than
Vmin, which means that any measurement of E for a particle in a
stationary or non-stationary state must give a result that
satisfies the inequality E > Vmin.
A1.1.3 QUANTUM MECHANICS OF MANY-PARTICLE SYSTEMSA1.1.3.1 THE
HYDROGEN ATOM
It is admittedly inconsistent to begin a section on
many-particle quantum mechanics by discussing a problem that can be
treated as a single particle. However, the hydrogen atom and atomic
ions in which only one electron remains (He+, Li2+ etc) are the
only atoms for which exact analytic solutions to the Schrdinger
equation can be obtained. In no cases are exact solutions possible
for molecules, even after the Born Oppenheimer approximation (see
section B3.1.1.1) is made to allow for separate treatment of
electrons and nuclei. Despite the limited interest of hydrogen
atoms and hydrogen-like ions to chemistry, the quantum mechanics of
these systems is both highly instructive and provides a basis for
treatments of more complex atoms and molecules. Comprehensive
discussions of one-electron atoms can be found in many textbooks;
the emphasis here is on qualitative aspects of the solutions. The
Schrdinger equation for a one-electron atom with nuclear charge Z
is
(A1.1.62)
where is the reduced mass of the electronnucleus system and the
Laplacian is most conveniently expressed in spherical polar
coordinates. While not trivial, this differential equation can be
solved analytically. Some of the solutions are normalizable, and
others are not. The former are those that describe the bound states
of oneelectron atoms, and can be written in the form(A1.1.63)
where N is a normalization constant, and Rnll(r) and Yl,m (, )
are specific functions that depend on the quantum numbers n, l and
ml. The first of these is called the principal quantum number,
while l is known as the angular momentum, or azimuthal, quantum
number, and ml the magnetic quantum number. The quantum numbers
that allow for normalizable wavefunctions are limited to integers
that run over the ranges
-24-
(A1.1.64) (A1.1.65) (A1.1.66)
The fact that there is no restriction on n apart from being a
positive integer means that there are an infinite number of
bound-state solutions to the hydrogen atom, a peculiarity that is
due to the form of the Coulomb potential. Unlike most bound state
problems, the range of the potential is infinite (it goes to zero
at large r, but diverges to negative infinity at r = 0). The
eigenvalues of the Hamiltonian depend only on the principal
quantum number and are (in attojoules (1018 J))
(A1.1.67)
where it should be noted that the zero of energy corresponds to
infinite separation of the particles. For each value of n, the
Schrdinger equation predicts that all states are degenerate,
regardless of the choice of l and ml. Hence, any linear combination
of wavefunctions corresponding to some specific value of n is also
an eigenfunction of the Hamiltonian with eigenvalue En. States of
hydrogen are usually characterized as ns, np, nd etc where n is the
principal quantum number and s is associated with l = 0, p with l =
1 and so on. The functions Rnl(r) describe the radial part of the
wavefunctions and can all be written in the form(A1.1.68)
where is proportional to the electronnucleus separation r and
the atomic number Z. Lnl is a polynomial of order n l 1 that has
zeros (where the wavefunction, and therefore the probability of
finding the electron, vanishesa radial node) only for positive
values of . The functions Yl,ml(, ) are the spherical harmonics.
The first few members of this series are familiar to everyone who
has studied physical chemistry: Y00 is a constant, leading to a
spherically symmetric wavefunction, while Y1,0, and specific linear
combinations of Y1,1 and Y1,1, vanish (have an angular node) in the
xy, xz and yz planes, respectively. In general, these functions
exhibit l nodes, meaning that the number of overall nodes
corresponding to a particular nlml is equal to n 1. For example,
the 4d state has two angular nodes (l = 2) and one radial node
(Lnl() has one zero for positive ). In passing, it should be noted
that many of the ubiquitous qualitative features of quantum
mechanics are illustrated by the wavefunctions and energy levels of
the hydrogen atom. First, the system has a zero-point energy,
meaning that the ground-state energy is larger than the lowest
value of the potential () and the spacing between the energy levels
decreases with increasing energy. Second, the ground state of the
system is nodeless (the electron may be found at any point in
space), while the number of nodes exhibited by the excited states
increases with energy. Finally, there is a finite probability that
the electron is found in a classically forbidden region in all
bound states. For the hydrogen atom ground state, this corresponds
to all electronproton separations larger than 105.8 pm, where the
electron is found 23.8% of the time. As usual, this tunnelling
phenomenon is less pronounced in excited states: the corresponding
values for the 3s state are 1904 pm and 16.0%. The Hamiltonian
commutes with the angular momentum operator z as well as that for
the square of the angular momentum 2. The wavefunctions above are
also eigenfunctions of these operators, with eigenvalues and . It
should be emphasized that the total angular momentum is , and not a
simple
-25-
integral multiple of as assumed in the Bohr model. In
particular, the ground state of hydrogen has zero angular momentum,
while the Bohr atom ground state has L = . The meaning associated
with the ml quantum number is more difficult to grasp. The choice
of z instead of x or y seems to be (and is) arbitrary and it is
illogical that a specific value of the angular momentum projection
along one coordinate must be observed in any experiment, while
those associated with x and y are not similarly restricted.
However, the states with a given l are degenerate, and the
wavefunction at any particular time will in general be some linear
combination of the ml eigenfunctions. The only way to isolate a
specific nlml (and therefore ensure the result of measuring Lz) is
to apply a magnetic field that lifts the degeneracy and breaks the
symmetry of the problem. The z axis
then corresponds to the magnetic field direction, and it is the
projection of the angular momentum vector on this axis that must be
equal to ml . The quantum-mechanical treatment of hydrogen outlined
above does not provide a completely satisfactory description of the
atomic spectrum, even in the absence of a magnetic field.
Relativistic effects cause both a scalar shifting in all energy
levels as well as splittings caused by the magnetic fields
associated with both motion and intrinsic properties of the charges
within the atom. The features of this fine structure in the energy
spectrum were successfully (and miraculously, given that it
preceded modern quantum mechanics by a decade and was based on a
two-dimensional picture of the hydrogen atom) predicted by a
formula developed by Sommerfeld in 1915. These interactions, while
small for hydrogen, become very large indeed for larger atoms where
very strong electronnucleus attractive potentials cause electrons
to move at velocities close to the speed of light. In these cases,
quantitative calculations are extremely difficult and even the
separability of orbital and intrinsic angular momenta breaks
down.A1.1.3.2 THE INDEPENDENT-PARTICLE APPROXIMATION
Applications of quantum mechanics to chemistry invariably deal
with systems (atoms and molecules) that contain more than one
particle. Apart from the hydrogen atom, the stationary-state
energies cannot be calculated exactly, and compromises must be made
in order to estimate them. Perhaps the most useful and widely used
approximation in chemistry is the independent-particle
approximation, which can take several forms. Common to all of these
is the assumption that the Hamiltonian operator for a system
consisting of n particles is approximated by the sum
(A1.1.69)
where the single-particle Hamiltonians i consist of the kinetic
energy operator plus a potential ( ) that does not explicitly
depend on the coordinates of the other n 1 particles in the system.
Of course, the simplest realization of this model is to completely
neglect forces due to the other particles, but this is often too
severe an approximation to be useful. In any event, the quantum
mechanics of a system described by a Hamiltonian of the form given
by equation (A1.1.69) is worthy of discussion simply because the
independent-particle approximation is the foundation for molecular
orbital theory, which is the central paradigm of descriptive
chemistry. Let the orthonormal functions i(1), j(2), . . ., p(n) be
selected eigenfunctions of the corresponding singleparticle
Hamiltonians 1, 2, . . ., n, with eigenvalues i, j, . . ., p. It is
easily verified that the product of these single-particle
wavefunctions (which are often called orbitals when the particles
are electrons in atoms and molecules)
-26(A1.1.70)
satisfies the approximate Schrdinger equation for the system
(A1.1.71)
with the corresponding energy
(A1.1.72)
Hence, if the Hamiltonian can be written as a sum of terms that
individually depend only on the coordinates of one of the particles
in the system, then the wavefunction of the system can be written
as a product of functions, each of which is an eigenfunction of one
of the single-particle Hamiltonians, hi. The corresponding
eigenvalue is then given by the sum of eigenvalues associated with
each single-particle wavefunction appearing in the product. The
approximation embodied by equation (A1.1.69), equation (A1.1.70),
equation (A1.1.71) and equation (A1.1.72) presents a conceptually
appealing picture of many-particle systems. The behaviour and
energetics of each particle can be determined from a simple
function of three coordinates and the eigenvalue of a differential
equation considerably simpler than the one that explicitly accounts
for all interactions. It is precisely this simplification that is
invoked in qualitative interpretations of chemical phenomena such
as the inert nature of noble gases and the strongly reducing
property of the alkali metals. The price paid is that the model is
only approximate, meaning that properties predicted from it (for
example, absolute ionization potentials rather than just trends
within the periodic table) are not as accurate as one might like.
However, as will be demonstrated in the latter parts of this
section, a carefully chosen independent-particle description of a
many-particle system provides a starting point for performing more
accurate calculations. It should be mentioned that even qualitative
features might be predicted incorrectly by independent-particle
models in extreme cases. One should always be aware of this
possibility and the oft-misunderstood fact that there really is no
such thing as an orbital. Fortunately, however, it turns out that
qualitative errors are uncommon for electronic properties of atoms
and molecules when the best independent-particle models are used.
One important feature of many-particle systems has been neglected
in the preceding discussion. Identical particles in quantum
mechanics must be indistinguishable, which implies that the exact
wavefunctions which describe them must satisfy certain symmetry
properties. In particular, interchanging the coordinates of any two
particles in the mathematical form of the wavefunction cannot lead
to a different prediction of the system properties. Since any
rearrangement of particle coordinates can be achieved by successive
pairwise permutations, it is sufficient to consider the case of a
single permutation in analysing the symmetry properties that
wavefunctions must obey. In the following, it will be assumed that
the wavefunction is real. This is not restrictive, as stationary
state wavefunctions for isolated atoms and molecules can always be
written in this way. If the operator Pij is that which permutes the
coordinates of particles i and j, then indistinguishability
requires that
(A1.1.73)
-27-
for any operator (including the identity) and choice of i and j.
Clearly, a wavefunction that is symmetric with respect to the
interchange of coordinates for any two particles(A1.1.74)
satisfies the indistinguishability criterion. However, equation
(A1.1.73) is also satisfied if the permutation of particle
coordinates results in an overall sign change of the wavefunction,
i.e.(A1.1.75)
Without further considerations, the only acceptable real
quantum-mechanical wavefunctions for an n-particle system would
appear to be those for which(A1.1.76)
where i and j are any pair of identical particles. For example,
if the system comprises two protons, a neutron and two electrons,
the relevant permutations are that which interchanges the proton
coordinates and that which interchanges the electron coordinates.
The other possible pairs involve distinct particles and the action
of the corresponding Pij operators on the wavefunction will in
general result in something quite different. Since
indistinguishability is a necessary property of exact
wavefunctions, it is reasonable to impose the same constraint on
the approximate wavefunctions formed from products of
single-particle solutions. However, if two or more of the i in the
product are different, it is necessary to form linear combinations
if the condition Pij = is to be met. An additional consequence of
indistinguishability is that the hi operators corresponding to
identical particles must also be identical and therefore have
precisely the same eigenfunctions. It should be noted that there is
nothing mysterious about this perfectly reasonable restriction
placed on the mathematical form of wavefunctions. For the sake of
simplicity, consider a system of two electrons for which the
corresponding single-particle states are i, j, k, . . ., n, with
eigenvalues i, j, k, . . ., n. Clearly, the two-electron
wavefunction = i (1)i(2) satisfies the indistinguishability
criterion and describes a stationary state with energy E0 = 2i.
However, the state i(1)j(2) is not satisfactory. While it is a
solution to the Schrdinger equation, it is neither symmetric nor
antisymmetric with respect to particle interchange. However, two
such states can be formed by taking the linear
combinations(A1.1.77) (A1.1.78)
which are symmetric and antisymmetric with respect to particle
interchange, respectively. Because the functions are orthonormal,
the energies calculated from S and A are the same as that
corresponding to the unsymmetrized product state i(1)j(2), as
demonstrated explicitly for S:
-28-
(A1.1.79)
It should be mentioned that the single-particle Hamiltonians in
general have an infinite number of solutions, so that an
uncountable number of wavefunctions can be generated from them.
Very often, interest is focused on the ground state of
many-particle systems. Within the independent-particle
approximation, this state can be represented by simply assigning
each particle to the lowest-lying energy level. If a calculation
is
performed on the lithium atom in which interelectronic repulsion
is ignored completely, the single-particle Schrdinger equations are
precisely the same as those for the hydrogen atom, apart from the
difference in nuclear charge. The following lithium atom
wavefunction could then be constructed from single-particle
orbitals(A1.1.80)
a form that is obviously symmetric with respect to interchange
of particle coordinates. If this wavefunction is used to calculate
the expectation value of the energy using the exact Hamiltonian
(which includes the explicit electronelectron repulsion
terms),(A1.1.81)
one obtains an energy lower than the actual result, which (see
(A1.1.4.1)) suggests that there are serious problems with this form
of the wavefunction. Moreover, a relatively simple analysis shows
that ionization potentials of atoms would increase
monotonicallyapproximately linearly for small atoms and
quadratically for large atomsif the independent-particle picture
discussed thus far has any validity. Using a relatively simple
model that assumes that the lowest lying orbital is a simple
exponential, ionization potentials of 13.6, 23.1, 33.7 and 45.5
electron volts (eV) are predicted for hydrogen, helium, lithium and
beryllium, respectively. The value for hydrogen (a one-electron
system) is exact and that for helium is in relatively good
agreement with the experimental value of 24.8 eV. However, the
other values are well above the actual ionization energies of Li
and Be (5.4 and 9.3 eV, respectively), both of which are smaller
than those of H and He! All freshman chemistry students learn that
ionization potentials do not increase monotonically with atomic
number, and that there are in fact many pronounced and more subtle
decreases that appear when this property is plotted as a function
of atomic number. There is evidently a grave problem here. The
wavefunction proposed above for the lithium atom contains all of
the particle coordinates, adheres to the boundary conditions (it
decays to zero when the particles are removed to infinity) and
obeys the restrictions P12 = P13 = P23 = that govern the behaviour
of the exact wavefunctions. Therefore, if no other restrictions are
placed on the wavefunctions of multiparticle systems, the product
wavefunction for lithium
-29-
must lie in the space spanned by the exact wavefunctions.
However, it clearly does not, because it is proven in subsection
(A1.1.4.1) that any function expressible as a linear combination of
Hamiltonian eigenfunctions cannot have an energy lower than that of
the exact ground state. This means that there is at least one
additional symmetry obeyed by all of the exact wavefunctions that
is not satisfied by the product form given for lithium in equation
(A1.1.80). This missing symmetry provided a great puzzle to
theorists in the early part days of quantum mechanics. Taken
together, ionization potentials of the first four elements in the
periodic table indicate that wavefunctions which assign two
electrons to the same single-particle functions such as
(A1.1.82)
(helium) and
(A1.1.83)
(beryllium, the operator
produces the labelled aabb product that is symmetric with
respect to
interchange of particle indices) are somehow acceptable but that
those involving three or more electrons in one state are not! The
resolution of this zweideutigkeit (two-valuedness) puzzle was made
possible only by the discovery of electron spin, which is discussed
below.A1.1.3.3 SPIN AND THE PAULI PRINCIPLE
In the early 1920s, spectroscopic experiments on the hydrogen
atom revealed a striking inconsistency with the Bohr model, as
adapted by Sommerfeld to account for relativistic effects. Studies
of the fine structure associated with the n = 4 n = 3 transition
revealed five distinct peaks, while six were expected from
arguments based on the theory of interaction between matter and
radiation. The problem was ultimately reconciled by Uhlenbeck and
Goudsmit, who reinterpreted one of the quantum numbers appearing in
Sommerfelds fine structure formula based on a startling assertion
that the electron has an intrinsic angular momentum independent of
that associated with its motion. This idea was also supported by
previous experiments of Stern and Gerlach, and is now known as
electron spin. Spin is a mysterious phenomenon with a rather
unfortunate name. Electrons are fundamental particles, and it is no
more appropriate to think of them as charges that resemble
extremely small billiard balls than as waves. Although they exhibit
behaviour characteristic of both, they are in fact neither.
Elementary textbooks often depict spin in terms of spherical
electrons whirling about their axis (a compelling idea in many
ways, since it reinforces the Bohr model by introducing a spinning
planet), but this is a purely classical perspective on electron
spin that should not be taken literally. Electrons and most other
fundamental particles have two distinct spin wavefunctions that are
degenerate in the absence of an external magnetic field. Associated
with these are two abstract states which are eigenfunctions of the
intrinsic spin angular momentum operator z(A1.1.84)
-30-
The allowed quantum numbers ms are and , and the corresponding
eigenfunctions are usually written as and , respectively. The
associated eigenvalues and give the projection of the intrinsic
angular momentum vector along the direction of a magnetic field
that can be applied to resolve the degeneracy. The overall spin
angular momentum of the electron is given in terms of the quantum
number s by . For an electron, s = . For a collection of particles,
the overall spin and its projection are given in terms of the spin
quantum numbers S and MS (which are equal to the corresponding
lower-case quantities for single particles) by and , respectively.
S must be positive and can assume either integral or half-integral
values, and the MS quantum numbers lie in the interval
(A1.1.85)
where a correspondence to the properties of orbital angular
momentum should be noted. The multiplicity of a state is given by
2S + 1 (the number of possible MS values) and it is customary to
associate the terms singlet with S = 0, doublet with , triplet with
S = 1 and so on. In the non-relativistic quantum mechanics
discussed in this chapter, spin does not appear naturally.
Although
Dirac showed in 1928 that a fourth quantum number associated
with intrinsic angular momentum appears in a relativistic treatment
of the free electron, it is customary to treat spin heuristically.
In general, the wavefunction of an electron is written as the
product of the usual spatial part (which corresponds to a solution
of the non-relativistic Schrdinger equation and involves only the
Cartesian coordinates of the particle) and a spin part , where is
either or . A common shorthand notation is often used,
whereby(A1.1.86) (A1.1.87)
In the context of electronic structure theory, the composite
functions above are often referred to as spin orbitals. When spin
is taken into account, one finds that the ground state of the
hydrogen atom is actually doubly degenerate. The spatial part of
the wavefunction is the Schrdinger equation solution discussed in
section (A1.1.3.1), but the possibility of either spin or means
that there are two distinct overall wavefunctions. The same may be
said for any of the excited states of hydrogen (all of which are,
however, already degenerate in the nonrelativistic theory), as the
level of degeneracy is doubled by the introduction of spin. Spin
may be thought of as a fourth coordinate associated with each
particle. Unlike Cartesian coordinates, for which there is a
continuous distribution of possible values, there are only two
possible values of the spin coordinate available to each particle.
This has important consequences for our discussion of
indistinguishability and symmetry properties of the wavefunction,
as the concept of coordinate permutation must be amended to include
the spin variable of the particles. As an example, the
independent-particle ground state of the helium atom based on
hydrogenic wavefunctions(A1.1.88)
must be replaced by the four possibilities(A1.1.89)
-31-
(A1.1.90) (A1.1.91) (A1.1.92)
While the first and fourth of these are symmetric with respect
to particle interchange and thereby satisfy the
indistinguishability criterion, the other two are not and
appropriate linear combinations must be formed. Doing so, one finds
the following four wavefunctions(A1.1.93) (A1.1.94) (A1.1.95)
(A1.1.96)
where the first three are symmetric with respect to particle
interchange and the last is antisymmetric. This suggests that under
the influence of a magnetic field, the ground state of helium might
be resolved into components that differ in terms of overall spin,
but this is not observed. For the lithium example, there are
eight possible ways of assigning the spin coordinates, only two
of which(A1.1.97) (A1.1.98)
satisfy the criterion Pij = . The other six must be mixed in
appropriate linear combinations. However, there is an important
difference between lithium and helium. In the former case, all
assignments of the spin variable to the state given by equation
(A1.1.88) produce a product function in which the same state (in
terms of both spatial and spin coordinates) appears at least twice.
A little reflection shows that it is not possible to generate a
linear combination of such functions that is antisymmetric with
respect to all possible interchanges; only symmetric combinations
such as(A1.1.99)
can be constructed. The fact that antisymmetric combinations
appear for helium (where the independentparticle ground state made
up of hydrogen 1s functions is qualitatively consistent with
experiment) and not for lithium (where it is not) raises the
interesting possibility that the exact wavefunction satisfies a
condition more restrictive than Pij = , namely Pij = . For reasons
that are not at all obvious, or even intuitive, nature does indeed
enforce this restriction, which is one statement of the Pauli
exclusion principle. When this idea is first met with, one usually
learns an equivalent but less general statement that applies only
within the independent-particle approximation: no two electrons can
have the same quantum numbers. What does this mean? Within the
independent-particle picture of an atom, each single-particle
wavefunction, or orbital, is described by the quantum numbers n, l,
ml and (when spin is considered) ms.
-32-
Since it is not possible to generate antisymmetric combinations
of products if the same spin orbital appears twice in each term, it
follows that states which assign the same set of four quantum
numbers twice cannot possibly satisfy the requirement Pij = , so
this statement of the exclusion principle is consistent with the
more general symmetry requirement. An even more general statement
of the exclusion principle, which can be regarded as an additional
postulate of quantum mechanics, is The wavefunction of a system
must be antisymmetric with respect to interchange of the
coordinates of identical particles and if they are fermions, and
symmetric with respect to interchange of and if they are bosons.
Electrons, protons and neutrons and all other particles that have s
= are known as fermions. Other particles are restricted to s = 0 or
1 and are known as bosons. There are thus profound differences in
the quantummechanical properties of fermions and bosons, which have
important implications in fields ranging from statistical mechanics
to spectroscopic selection rules. It can be shown that the spin
quantum number S associated with an even number of fermions must be
integral, while that for an odd number of them must be
half-integral. The resulting composite particles behave
collectively like bosons and fermions, respectively, so the
wavefunction symmetry properties associated with bosons can be
relevant in chemical physics. One prominent example is the
treatment of nuclei, which are typically considered as composite
particles rather than interacting protons and neutrons. Nuclei with
even atomic number therefore behave like individual bosons and
those with odd atomic number as fermions, a distinction that plays
an important role in rotational spectroscopy of polyatomic
molecules.
A1.1.3.4 INDEPENDENT-PARTICLE MODELS IN ELECTRONIC STRUCTURE
At this point, it is appropriate to make some comments on the
construction of approximate wavefunctions for the many-electron
problems associated with atoms and molecules. The Hamiltonian
operator for a molecule is given by the general form
(A1.1.100)
It should be noted that nuclei and electrons are treated
equivalently in , which is clearly inconsistent with the way that
we tend to think about them. Our understanding of chemical
processes is strongly rooted in the concept of a potential energy
surface which determines the forces that act upon the nuclei. The
potential energy surface governs all behaviour associated with
nuclear motion, such as vibrational frequencies, mean and
equilibrium internuclear separations and preferences for specific
conformations in molecules as complex as proteins and nucleic
acids. In addition, the potential energy surface provides the
transition state and activation energy concepts that are at the
heart of the theory of chemical reactions. Electronic motion,
however, is never discussed in these terms. All of the important
and useful ideas discussed above derive from the BornOppenheimer
approximation, which is discussed in some detail in section B3.1.
Within this model, the electronic states are solutions to the
equation
-33-
(A1.1.101)
where